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The height of a polygon is the length of a perpendicular from the base to the vertex most distant from the base.

In the words of Euclid:

The height of any figure is the perpendicular drawn from the vertex to the base.

(The Elements: Book $\text{VI}$: Definition $4$)

Also known as

The height of a polygon is also known as its altitude.

However, on $\mathsf{Pr} \infty \mathsf{fWiki}$ it is preferred that the term altitude be reserved for the actual perpendicular line itself, and for the term height to be used for its length.

Also see